Suppression of blow up by a logistic source in $2$D Keller-Segel system with fractional dissipation

TL;DR

This paper studies a 2D Keller-Segel model with fractional diffusion and logistic growth, proving global existence of solutions under certain conditions, which helps prevent blow-up phenomena.

Contribution

It establishes the existence of global regular and weak solutions for the Keller-Segel system with fractional dissipation and logistic source, extending previous results to broader parameter ranges.

Findings

Global regular solutions exist for certain fractional orders and initial data.

Weak solutions are obtainable for a wider range of fractional orders.

The logistic term suppresses blow-up in the Keller-Segel system.

Abstract

We consider a two dimensional parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order $\alpha$. We obtain existence of global in time regular solution for arbitrary initial data with no size restrictions and $c<\alpha\leq 2$, where $c \in (0,2)$ depends on the equation's parameters. For an even wider range of $\alpha's$, we prove existence of global in time weak solution for general initial data.